2010 Mathematics subject classification: primary 93C30; secondary 93A30, 68W30, 92C45, 93B99.Keywords and phrases: linear switching systems, global a priori identifiability, compartmental modelling, systems theory, symbolic algebra, biosensors, biochemical kinetics, kinetic experiments.Ideally, a parametric model for a biological system enables prediction of system behaviour for conditions where we lack observations. This necessitates first estimating parameters from some limited data series subject to random error, that is, solving an 'inverse problem'. A solution is some parameter vector that optimises an objective function. For example, a solution may minimise a sum of squared errors. Multiple (equally valid) solutions may result in unresolvable uncertainty over which is the actual parameter vector. This is problematic as predictions for a system's observable features-and the unobservable 'state variables' influencing these-may vary drastically with the parameter vector employed. Hence, we cannot confidently use our model to predict system behaviour. Consequently, the effort and resources expended in collecting data provide no benefit.We may anticipate this problem prior to data collection. We achieve this by testing the combination of a model and proposed experimental conditions for the property of (global a priori) identifiability. Testing occurs in an idealised setting which assumes that an infinite, error-free data record is available. It determines those parameter vectors for which model output exactly reproduces such 'idealised data'.Commonly, errors do not provide information on the model parameters. In this case, it is almost certain that the inverse problem's solution set cannot be smaller than that found by the identifiability test. That is, if the test returns uncountably infinitely many solutions, we are almost guaranteed an uninformative study. A test returning a unique solution shows the diametrically opposite outcome; it is at least possible for proposed experiments to yield a decisive result. Our interest in identifiability pertains to the modelling of flow-cell optical biosensor experiments. These indirectly monitor the formation and dissociation of complexes of biochemical species. Experimentalists use data with an assumed model for the estimation of parameters representing rate constants.Often experiments have multiple stages, delineated by an abrupt change in experimental conditions. Accordingly, in certain situations, experimental data is suitably modelled by a type of linear switching system (LSS). As experiments indirectly measure the transfer of mass between forms, and this mass is conserved, suitable models are also 'compartmental'. There is a scant literature on testing LSSs for identifiability, in particular for those which evolve in continuous-time.Our application leads us to focus on the analysis of continuous-time uncontrolled compartmental LSS of one switching event (ULSS-1). These may suitably model data from a common ('kinetic') type of biosensor experiment having two stages. ...
Flow-cell optical biosensors are a popular means of studying biomolecular interactions. The time course of data produced shows the progress of the interaction in real time. In a quantitative study, data is used to estimate parameters, such as rate constants, in a mathematical model of interactant concentrations over time. A study unable to assign a unique estimate to each parameter may be less informative than desired. This result can be anticipated prior to data collection by testing the assumed model class for global a priori identifiability. In the case where an interaction model is an uncontrolled continuous-time linear switching system, the testing method available is applicable only in a special case. This paper proposes an algorithm that is sufficient for classifying a previously unclassified test case as globally a priori identifiable.Keywords-Flow-cell optical biosensor experiments; biomolecular interaction; global a priori identifiability; experimental design.
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